This laptop has been running 24/7 for the last 3 years or such, so it's not too surprising that a disk dies. Surprisingly though, for the first time in a long series of dead disks, smartctl -a does indeed show errors for this disk. Here's a short snippet of those:

The [_U] means that one of two disks has failed, it should normally be [UU]. There are two RAID-1s actually, a small md0 (sda6 + sdb6) for /boot and the main md1 (sda7 + sdb7) which holds the OS and my data. Apparently (at first at least), only sda7 was faulty and got kicked out of the array:

$ dmesg | grep kick
md: kicking non-fresh sda7 from array!

Anyway, so I ordered a replacement disk, removed the dead disk (I checked the serial number and brand before, so I don't accidentally remove the wrong one), inserted the new disk and rebooted.

Note: In order for this to work you have to have (previously) installed the bootloader (usually GRUB) onto both disks, otherwise you won't be able to boot from either of them (which you'll want to do if one of them dies, of course). In my case, sda was now dead, so I put sdb into its place (physically, by using the other SATA connector/port) and the new replacement disk would become the new sdb.

After the reboot, the new disk needs to be partitioned like the other RAID disk. This can be done easily by copying the partition layout of the "good" disk (now sda after the reboot) onto the empty disk (sdb):

$ sfdisk -d /dev/sda | sfdisk /dev/sdb

Specifically, the RAID disks/partitions need to have the type/ID "fd" ("Linux raid autodetect"), check if that is the case. Then, you can add the new disk to the RAIDs:

$ mdadm /dev/md0 --add /dev/sdb6
$ mdadm /dev/md1 --add /dev/sdb7

After a few hours the RAID will be re-synced properly and all is good again. You can check the progress via:

$ watch -n 1 cat /proc/mdstat

You should probably not reboot during the resync (though I'm not 100% sure if that would be an issue in practice; please leave a comment if you know).

Also, don't forget to install GRUB on the new disk so you can still boot when the next disk dies:

$ grub-mkdevicemap
$ grub-install /dev/sdb

And it might be a good idea to use S.M.A.R.T. to check the new disk, just in case. I did a quick run for the new disk via:

Restarting during an mdadm resync is not fatal. However, until the resync finishes, you are at the same risk of data loss you would have without the RAID getting the new disk in. Since many disk failures occur when power is cycled, there is a slightly increased chance for getting things messed up by restarting during the resync.

Disk activity during the resync slows it down considerably; this is especially evident in RAID levels other than RAID1/10 and with larger volumes.